Lecture 3 b Central It y

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degree (numberof connections)denoted by size

closeness(length of

shortest path toall others)denoted by color

Closeness and Lada's fb network

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Eigenvector centrality

How central you are depends on how

central your neighbors are

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c(b )= a (I- b A)- 1

A1a is a normalization constant

b determines how important the centrality of your neighborsis

Ais the adjacency matrix (can be weighted)

Iis the identity matrix (1s down the diagonal, 0 off-diagonal)

1is a matrix of all ones.

Bonacich eigenvector centrality

ci(b ) = (a + b cjj

)Aji

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small b high attenuation

only your immediate friends matter, andtheir importance is factored in only a bit

high b low attenuationglobal network structure matters (your

friends, your friends' of friends etc.)

= 0 yields simple degree centrality

Bonacich Power Centrality: attenuation factorb

ci(b ) = (a + b cjj

)Aji

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Ifb > 0, nodes have higher centrality when they have

edges to other central nodes.

Ifb < 0, nodes have higher centrality when they have

edges to less central nodes.

Bonacich Power Centrality: attenuation factorb

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b=.25

b=-.25

Why does the middle node have lower centrality than its

neighbors when b is negative?

Bonacich Power Centrality: examples

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Centrality in directed networks

WWW

food webs

population dynamics influence

hereditary

citation

transcription regulation networks

neural networks

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Betweenness centrality in directed networks

We now consider the fraction of all directed paths

between any two vertices that pass through a node

Only modification: when normalizing, we have(N-1)*(N-2) instead of (N-1)*(N-2)/2, because wehave twice as many ordered pairs as unorderedpairs

CB (i) = gjkj,k

(i) /gjk

betweenness of vertex ipaths between j and k that pass through i

all paths between j and k

CB

'(i) =C

B

(i)/[(N - 1)(N - 2)]

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Directed geodesics

A node does not necessarily lie on a geodesic

fromjto kif it lies on a geodesic from ktoj

k

j

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Directed closeness centrality

choose a direction

in-closeness (e.g. prestige in citation networks)

out-closeness

usually consider only vertices from which thenode iin question can be reached

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Eigenvector centrality in directednetworks

PageRank brings order to the Web:

it's not just the pages that point to you, buthow many pages point to those pages, etc.

more difficult to artificially inflate centralitywith a recursive definition

Many webpages scattered

across the web

an important page, e.g. slashdot

if a web page is

slashdotted, it gains attention

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Ranking pages by tracking a drunk

A random walker following

edges in a network for a very

long time will spend a

proportion of time at each

node which can be used asa measure of importance

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Trapping a drunk

Problem with pure random walk metric:

Drunk can be trapped and end up going in circles

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Ingenuity of the PageRank algorithm

Allow drunk to teleport with some probability

e.g. random websurfer follows links for a while, but with

some probability teleports to a random page

(bookmarked page or uses a search engine to start anew)

l b bl l ti f

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example: probable location ofrandom walker after 1 step

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Pag

eRank

t=1

20% teleportation probability

slide adapted from: Dragomir Radev

l b bl l ti f

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1

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PageRank

t=10

slide from: Dragomir Radev

example: probable location ofrandom walker after 10 steps

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GUESS PageRank demo

Quiz Q:

What happens to the

relative PageRank scoresof the nodes as you

increase the teleportation

probability?

they equalize they diverge

they are unchanged

http://www.ladamic.com/netlearn/GUESS/pagerank.html

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wrap up

Centralitymany measures: degree, betweenness,

closeness, eigenvector

may be unevenly distributed

measure via distributions and centralization

in directed networks

indegree, outdegree, PageRank

consequences:

benefits & risks (Baker & Faulkner)

information flow & productivity (Aral & VanAlstyne)